# Preserver problems for the logics associated to Hilbert spaces and related Grassmannians

## TL;DR

This paper explores transformations preserving orthogonality and compatibility in the quantum logic of Hilbert spaces, extending classical results like Wigner's theorem to Grassmannians and related structures.

## Contribution

It provides new insights into preservers of orthogonality and compatibility relations in quantum logics and Grassmannians, generalizing classical theorems.

## Key findings

- Characterization of transformations preserving orthogonality
- Results on compatibility relation preservers
- Extensions of Wigner's theorem to Grassmannians

## Abstract

We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.